Problem 49 - College Algebra and Calculus: An Applied Approach

Question

Use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. $$ f(x)=\sqrt{\frac{x+1}{x}} $$

Step-by-Step Solution

Answer

The derivative of \( f(x)= \sqrt{\frac{x+1}{x}} \) is \( f'(x)= -\frac{(3x+1)}{2 * (x+1)^{1/2} * x^{3/2}} \) The shape of the original function changes at the points where the derivative is zero.

Step 1: Finding the derivative

First, manipulate the function \( f(x)=\sqrt{\frac{x+1}{x}} \) to make it easier to differentiate. Rewrite it as \( f(x)= (x+1)^{1/2} * x^{-1/2} \). Now, apply the product rule which states that the derivative of a product of two functions is the derivative of the first times the second plus the first times the derivative of the second. Apply the power rule, which states that the derivative of \( x^n \) is \( n * x^{(n-1)} \). This results in the derivative, \( f'(x)= - \frac{1}{2} * (x+1)^{-1/2} * x^{-1/2} + (x+1)^{1/2} * (-1/2) * x^{-3/2} \). After simplifying we get \( f'(x)= - \frac{(3x+1)}{2 * (x+1)^{1/2} * x^{3/2}} \)

Step 2: Graphing the function and its derivative

Use a graphing utility to sketch both the original function \( \sqrt{\frac{x+1}{x}} \) and its derivative \( -\frac{(3x+1)}{2 * (x+1)^{1/2} * x^{3/2}} \). Make sure they are on the same graph for comparison.

Step 3: Studying the points where the derivative is zero

From the graph, identify the points where the derivative cuts the x-axis; these are the points where the derivative is zero. At these points, study the behavior of the original function. The function typically has a local maximum or minimum at these points. Again refer to the graph of the original function to confirm this.